Questions: A sales assistant is hanging 11 sweaters on a rack. He has 5 white sweaters, 4 green sweaters, and 2 pink sweaters. In how many distinct orders can the sweaters be arranged if two sweaters of the same color are considered identical (not distinct)?

A sales assistant is hanging 11 sweaters on a rack. He has 5 white sweaters, 4 green sweaters, and 2 pink sweaters. In how many distinct orders can the sweaters be arranged if two sweaters of the same color are considered identical (not distinct)?

Transcript text: A sales assistant is hanging 11 sweaters on a rack. He has 5 white sweaters, 4 green sweaters, and 2 pink sweaters. In how many distinct orders can the sweaters be arranged if two sweaters of the same color are considered identical (not distinct)? $\square$

Solution

Solution Steps

Step 1: Calculate the factorial of the total number of objects, \(n!\)

Given \(n = 11\), we find \(n! = 39916800\).

Step 2: Calculate the product of factorials of the number of identical objects, \(\prod_{i=1}^{k} n_i!\)

Given \(n_i\) values as [5, 4, 2], we calculate \(\prod_{i=1}^{k} n_i! = 5! \cdot 4! \cdot 2! = 5760\).

Step 3: Calculate the number of distinct arrangements

Using the formula \(\dfrac{n!}{\prod_{i=1}^{k} n_i!}\), we find the number of distinct arrangements to be \(\dfrac{39916800}{5760}\) = 6930.

Final Answer

The number of distinct arrangements of the objects is 6930.

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